# Whether convolution followed by down sampling equals wavelet decomposition

I know that wavelet decomposition equals convolution followed by down sampling by two.I tried to test this on mat lab on haar wavelet. I write the below code and compared with the default wavedec command in mat lab. The code I wrote is shown below.

clear all
clc
h=[0.7071    0.7071];% Haar scaling filter
g=[-0.7071    0.7071];%Haar wavelet filter
x=[1:10];% Input
h=fliplr(h);% h(-m)
g=fliplr(g);%g(-m)
s=conv(h,x);
s=downsample(s,2)
d=conv(g,x);
d=downsample(d,2)
[a L]=wavedec(x,1,'haar') % for comparison


After running this program I get s = 0.7071 3.5355 6.3639 9.1923 12.0207 7.0710. d = 0.7071 0.7071 0.7071 0.7071 0.7071 -7.0710 and the wavedec out put as a = 2.1213 4.9497 7.7782 10.6066 13.4350 -0.7071 -0.7071 -0.7071 -0.7071 -0.7071. L = 5 5 10

I dont know why wavedec and my code gives different out puts, will any one help???

• where did you read that, in which context? Aug 16, 2016 at 14:16
• In every context filtering corresponds to convolution. Aug 16, 2016 at 14:41
• true, but still, this is definitely (and trivially to counter-prove by convolving with $\delta(t)$) not the case for every filter. and hence, you're missing an important part of the original description. And hence, I ask for context. Aug 16, 2016 at 14:45
• In case of Discrete wavelet transform, high pass and low pass coefficients at each level are obtained with the convolution of the corresponding filter coefficients followed by down sampling.. for details please go to matlab website ,(following link and go to more about section) in.mathworks.com/help/wavelet/ref/wavedec.html Aug 16, 2016 at 14:53
• in.mathworks.com/help/wavelet/ref/wavedec.html Aug 16, 2016 at 15:01

I found out it, downsample command in matlab takes first element then 3rd,5th etc where downsampling in wavedec starts from 2,4,6...more over fliping of wavelet filters are not required ..Instead of using downsample command we can repalce with s=s(2:2:length(s))